Twisted filtrations of Soergel bimodules and linear Rouquier complexes

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in the case where the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin-Tits group. These lifts turn out to be generalizations of Mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups.